Grande Randonnée - What is the best type of workout?

I am going to be taking some time to hike in France this spring/summer period, and I am aiming for some of the Grande Randonnée (GR) trails that are all over. I prefer to look at mountains rather than climb every one, but I will be aiming on doing around 35km or 20 miles a day.

Currently, without trying hard, I can go about 5.5 kph, or 3.5 mph, and can maintain this for about 8 hours. That makes it around 28 km, but I expect the terrain to be rougher and the pack heavier. I have heard that you should walk for health and run to get in shape.

Should I be training:- All weights and no running?- Mainly Running and no walking long distances?- More walking than the other two?